Self-configuring extremum-seeking control system

ABSTRACT

A self-configuring extremum-seeking controller includes a dither signal generator, a communications interface, a phase delay estimator, and a bandwidth estimator. The dither signal generator identifies a stored dither frequency, generates a dither signal having the stored dither frequency, and uses the dither signal to perturb a control input for a plant. The communications interface provides the perturbed control input to the plant and receives an output signal from the plant resulting from the perturbed control input. The phase delay estimator estimates a phase delay between the output signal and the dither signal. The bandwidth estimator estimates a bandwidth of the plant based on the estimated phase delay. The dither signal generator updates the stored dither frequency based on the estimated bandwidth.

BACKGROUND

The present disclosure relates generally to extremum-seeking controlstrategies. The present disclosure relates more particularly toregulating, via extremum-seeking control, a variable of interest (e.g.,power production, power consumption, rate of refrigerant flow, etc.) inan energy system.

Extremum-seeking control (ESC) is a class of self-optimizing controlstrategies that can dynamically search for the unknown and/ortime-varying inputs of a system for optimizing a certain performanceindex. It can be considered a dynamic realization of gradient searchingthrough the use of dither signals. The gradient of the system outputwith respect to the system input is typically obtained by slightlyperturbing the system operation and applying a demodulation measure.Optimization of system performance can be obtained by driving thegradient towards zero by using an integrator in the closed-loop system.ESC is a non-model based control strategy, meaning that a model for thecontrolled system is not necessary for ESC to optimize the system. ESChas been used in many different engineering applications (e.g.,combustion, circuitry, mining, aerospace and land-based vehicles,building HVAC, wind and solar energy, etc.) and has been shown to beable to improve the operational efficiency and performance for theseengineering applications.

Although ESC does not require a model of the system to optimize thesystem output, properly configuring an ESC system may require knowledgeof the system bandwidth (e.g., the natural frequency ω_(n) of the plant)to select the frequency ω_(d) of the dither signal. For example, it maybe desirable to select a dither signal frequency ω_(d) that is the sameor similar to the natural frequency ω_(n) of the plant to enhance theeffect of the dither signal d on the performance variable y (e.g., byincreasing the resonance between the dither signal d and the plant).Additionally, it may be desirable to know the system gain K_(s) in orderto properly set the rate of gradient descent.

Traditional ESC systems require manual configuration and testing todetermine suitable values for system parameters such as the systembandwidth and system gain. For example, a manual step test systemidentification procedure can be performed to estimate the systembandwidth. However, such testing requires disturbing the system (causingoperation disruption) and assumes that the bandwidth does not changeafter testing. It would be desirable to provide an automaticconfiguration procedure that is non-disruptive and can automaticallydetermine system parameters without requiring manual testing andconfiguration.

SUMMARY

One implementation of the present disclosure is a self-configuringextremum-seeking controller. The controller includes a dither signalgenerator, a communications interface, a phase delay estimator, and abandwidth estimator. The dither signal generator identifies a storeddither frequency, generates a dither signal having the stored ditherfrequency, and uses the dither signal to perturb a control input for aplant. The communications interface provides the perturbed control inputto the plant and receives an output signal from the plant resulting fromthe perturbed control input. The phase delay estimator estimates a phasedelay between the output signal and the dither signal. The bandwidthestimator estimates a bandwidth of the plant based on the estimatedphase delay. The dither signal generator updates the stored ditherfrequency based on the estimated bandwidth.

In some embodiments, the dither signal generator generates a new dithersignal having the updated dither frequency and uses the new dithersignal to perturb the control input for the plant.

In some embodiments, the bandwidth estimator estimates a naturalfrequency of the plant based on the estimated phase delay and uses theestimated natural frequency of the plant as the estimated bandwidth. Insome embodiments, the bandwidth estimator estimates the bandwidth of theplant as a function of the estimated phase delay and the stored ditherfrequency.

In some embodiments, the phase delay estimator calculates a dot productof the dither signal and the output signal, calculates 2-norms of thedither signal and the output signal, and estimates the phase delay as afunction of the dot product and the 2-norms.

In some embodiments, the controller includes an exponentially-weightedmoving average (EWMA) calculator that calculates a first EWMA of thedither signal and a second EWMA of the output signal from the plant. Insome embodiments, the phase delay estimator estimates the phase delay asa function of the first and second EWMAs. In some embodiments, the gainestimator estimates the system gain as a function of the first andsecond EWMAs.

In some embodiments, the controller includes a gain estimator thatestimates a system gain based on the estimated bandwidth and the storeddither frequency. The controller may include an integrator that uses theestimated system gain to scale a step size of a gradient descentprocedure performed by the controller.

Another implementation of the present disclosure is a method forself-configuring one or more extremum-seeking control parameters used byan extremum-seeking controller to modulate a control input for a plant.The method includes receiving a stored dither frequency at theextremum-seeking controller, generating a dither signal having thestored dither frequency, and using the dither signal to perturb thecontrol input for the plant. The method includes providing the perturbedcontrol input from the controller to the plant and receiving an outputsignal from the plant resulting from the perturbed control input. Themethod includes estimating a phase delay between the output signal andthe dither signal and estimating a bandwidth of the plant based on theestimated phase delay. The method includes updating the stored ditherfrequency based on the estimated bandwidth. The generating, providing,estimating, and updating steps may be performed automatically by thecontroller.

In some embodiments, the method includes generating a new dither signalhaving the updated dither frequency and using the new dither signal toperturb the control input for the plant.

In some embodiments, estimating the bandwidth of the plant includesestimating a natural frequency of the plant based on the estimated phasedelay and using the estimated natural frequency of the plant as theestimated bandwidth. In some embodiments, the bandwidth of the plant isestimated as a function of the estimated phase delay and the storeddither frequency.

In some embodiments, estimating the phase delay between the outputsignal and the dither signal includes calculating a dot product of thedither signal and the output signal, calculating 2-norms of the dithersignal and the output signal, and estimating the phase delay as afunction of the dot product and the 2-norms.

In some embodiments, the method includes calculating a firstexponentially-weighted moving average (EWMA) of the dither signal and asecond EWMA of the output signal from the plant. In some embodiments,the phase delay is estimated as a function of the first and secondEWMAs. In some embodiments, the system gain is estimated as a functionof the first and second EWMAs.

In some embodiments, the method includes estimating a system gain basedon the estimated bandwidth and the stored dither frequency and using theestimated system gain to scale a step size of a gradient descentprocedure performed by the controller.

Another implementation of the present disclosure is a self-configuringextremum-seeking controller. The controller includes a dither signalgenerator, a communications interface, and a parameter estimator. Thedither signal generator generates a dither signal and uses the dithersignal to perturb a control input for a plant. The communicationsinterface provides the perturbed control input to the plant and receivesan output signal from the plant resulting from the perturbed controlinput. The parameter estimator uses the output signal from the plant inconjunction with the dither signal to generate values for one or moreextremum-seeking control parameters. The dither signal generator usesthe generated values for the extremum-seeking control parameters togenerate the dither signal.

In some embodiments, the parameter estimator includes a phase delayestimator that estimates a phase delay between the output signal and thedither signal and a bandwidth estimator that estimates a bandwidth ofthe plant based on the estimated phase delay. The dither signalgenerator may use the estimated bandwidth to update a stored ditherfrequency used to generate the dither signal.

In some embodiments, the parameter estimator includes a gain estimatorthat estimates a system gain based on the estimated bandwidth and thestored dither frequency. The controller may include an integrator thatuses the estimated system gain to scale a step size of a gradientdescent procedure performed by the controller.

In some embodiments, the controller includes an exponentially-weightedmoving average (EWMA) calculator that calculates a first EWMA of thedither signal and a second EWMA of the output signal from the plant. Atleast one of the phase delay and the system gain may be estimated as afunction of the first and second EWMAs.

Those skilled in the art will appreciate that the summary isillustrative only and is not intended to be in any way limiting. Otheraspects, inventive features, and advantages of the devices and/orprocesses described herein, as defined solely by the claims, will becomeapparent in the detailed description set forth herein and taken inconjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a drawing of a building equipped with a HVAC system in whichthe present invention may be implemented, according to an exemplaryembodiment.

FIG. 2 is a block diagram illustrating the HVAC system of FIG. 1 ingreater detail, according to an exemplary embodiment.

FIG. 3A is a block diagram of an example single-input single-output(SISO) extremum-seeking control (ESC) system in which the presentinvention may be implemented, according to an exemplary embodiment.

FIG. 3B is a block diagram of an example dither switching ESC system inwhich the present invention may be implemented, according to anexemplary embodiment.

FIG. 4 is a block diagram of another example ESC system in which thepresent invention may be implemented, according to an exemplaryembodiment.

FIG. 5 is a block diagram of a self-configuring extremum-seeking control(SCESC) system including a SCESC controller and a plant, according to anexemplary embodiment.

FIG. 6 is a flowchart of a process for automatically configuring an ESCsystem which may be performed by the SCESC controller of FIG. 5,according to an exemplary embodiment.

FIG. 7 is a flowchart of another process for automatically configuringan ESC system which may be performed by the SCESC controller of FIG. 5,according to an exemplary embodiment.

FIG. 8 is a pair of graphs illustrating two outputs of an automaticparameter estimation process (i.e., natural frequency ω_(n) and systemgain K_(s)) which may be performed by the SCESC controller of FIG. 5,according to an exemplary embodiment.

FIG. 9 is a graph of a dither signal which may be generated by the SCESCcontroller of FIG. 5, showing an adjustment to the dither signalfrequency as a result of the automatic parameter estimation process,according to an exemplary embodiment.

DETAILED DESCRIPTION Overview

Referring generally to the FIGURES, a self-configuring extremum-seekingcontrol (SCESC) system and components thereof are shown according tovarious exemplary embodiments. In general, extremum-seeking control(ESC) is a class of self-optimizing control strategies that candynamically search for the unknown and/or time-varying inputs of asystem for optimizing a certain performance index. It can be considereda dynamic realization of gradient searching through the use of dithersignals. The gradient of the system output with respect to the systeminput is typically obtained by slightly perturbing the system operationand applying a demodulation measure. Optimization of system performancecan be obtained by driving the gradient towards zero by using anintegrator in the closed-loop system. ESC is a non-model based controlstrategy, meaning that a model for the controlled system is notnecessary for ESC to optimize the system. Various implementations of ESCare described in detail in U.S. Pat. No. 8,473,080, U.S. Pat. No.7,827,813, U.S. Pat. No. 8,027,742, U.S. Pat. No. 8,200,345, U.S. Pat.No. 8,200,344, U.S. patent application Ser. No. 14/495,773, and U.S.patent application Ser. No. 14/538,700. The entire disclosure of each ofthese patents and patent applications is incorporated by referenceherein.

The SCESC system described herein includes a self-configuringextremum-seeking controller (SCESC) and a plant. The controller receivesfeedback from the plant and provides a control input to the plant. Thecontroller perturbs the control input u to the plant with a dithersignal d and analyzes the effect of such perturbation on an observedperformance variable y. The controller adjusts the control input u todrive the gradient of performance variable y to zero. In this way, thecontroller identifies values for control input u that achieve an optimalvalue (e.g., a maximum or a minimum) for performance variable y.

Although ESC does not require a model of the system to optimizeperformance variable y, properly configuring an ESC system may requireknowledge of the system bandwidth (e.g., the natural frequency ω_(n) ofthe plant) to select the frequency ω_(d) of the dither signal d. Forexample, it may be desirable to select a dither signal frequency ω_(d)that is the same or similar to the natural frequency ω_(n) of the plantto enhance the effect of the dither signal d on the performance variabley (e.g., by increasing the resonance between the dither signal d and theplant). Additionally, it may be desirable to know the system gain K_(s)in order to properly set the rate of gradient descent. Advantageously,the SCESC controller described herein may be configured to automaticallydetermine both the system bandwidth and the system gain using the dithersignal d to perform system identification.

In some embodiments, the SCESC controller determines the naturalfrequency ω_(n) of the plant and uses the natural frequency ω_(n) as thesystem bandwidth. The SCESC controller may determine the naturalfrequency ω_(n) based on the phase delay φ between the dither signal dand the performance variable y. For example, the SCESC controller mayexcite the plant with a dither signal d having a known dither frequencyω_(d) and may measure the resultant phase delay φ of the performancevariable y relative to the dither signal d. The SCESC controller may usethe phase delay φ in conjunction with the dither signal frequency ω_(d)to estimate the natural frequency ω_(n), as shown in the followingequation:

$\omega_{n} = {- {\omega_{d}\left( {{\cot (\varphi)} - \sqrt{\frac{1}{\sin^{2}(\varphi)}}} \right)}}$

The SCESC controller may use the natural frequency ω_(n) to set orupdate the dither signal frequency ω_(d) (e.g., by setting the dithersignal frequency ω_(d) equal to the natural frequency ω_(n)).

In some embodiments, the SCESC controller uses a history of values forthe dither signal d to calculate an exponentially-weighted movingaverage (EWMA) s_(d,k) of the dither signal d at time k. Similarly, theSCESC controller may use a history of values for the performancevariable y to calculate an EWMA s_(y,k) of the performance variable y attime k. The SCESC controller may use the EWMAs s_(d,k) and s_(y,k) inconjunction with the natural frequency ω_(n) and the known dither signalfrequency ω_(d) to estimate the system gain K_(s), as shown in thefollowing equation:

$K_{s} = \sqrt{\frac{s_{y,k}}{s_{d,k}}\left( {\left( {1 - \left( \frac{\omega_{d}}{\omega_{n}} \right)^{2}} \right)^{2} + \left( {2\zeta \frac{\omega_{d}}{\omega_{n}}} \right)^{2}} \right)}$

The SCESC controller may use the system gain K_(s) to set or update therate of gradient descent.

Advantageously, automatically determining the system bandwidth and thesystem gain enables the SCESC system to be configured automatically(e.g., without manual testing and configuration) and updatedperiodically as the ESC parameters and/or system conditions change.Additional features and advantages of the present invention aredescribed in greater detail below.

Building and HVAC System

Referring now to FIGS. 1-2, an exemplary building 10 and HVAC system 20in which an extremum-seeking control system may be implemented areshown, according to an exemplary embodiment. Although the ESC systemsand methods of the present disclosure are described primarily in thecontext of a building HVAC system, it should be understood that ESC maybe generally applicable to any type of control system that optimizes orregulates a variable of interest. For example, the ESC systems andmethods of the present disclosure may be used to optimize an amount ofenergy produced by various types of energy producing systems or devices(e.g., power plants, steam or wind turbines, solar panels, combustionsystems, etc.) and/or to optimize an amount of energy consumed byvarious types of energy consuming systems or devices (e.g., electroniccircuitry, mechanical equipment, aerospace and land-based vehicles,building equipment, HVAC devices, refrigeration systems, etc.).

In various implementations, ESC may be used in any type of controllerthat functions to achieve a setpoint for a variable of interest (e.g.,by minimizing a difference between a measured or calculated input and asetpoint) and/or optimize a variable of interest (e.g., maximize orminimize an output variable). It is contemplated that ESC can be readilyimplemented in various types of controllers (e.g., motor controllers,power controllers, fluid controllers, HVAC controllers, lightingcontrollers, chemical controllers, process controllers, etc.) andvarious types of control systems (e.g., closed-loop control systems,open-loop control systems, feedback control systems, feed-forwardcontrol systems, etc.) as may be suitable for various applications. Allsuch implementations should be considered within the scope of thepresent disclosure.

Referring particularly to FIG. 1, a perspective view of building 10 isshown. Building 10 is served by HVAC system 20. HVAC system 20 is shownto include a chiller 22, a boiler 24, a rooftop cooling unit 26, and aplurality of air-handling units (AHUs) 36. HVAC system 20 uses a fluidcirculation system to provide heating and/or cooling for building 10.The circulated fluid may be cooled in chiller 22 or heated in boiler 24,depending on whether cooling or heating is required. Boiler 24 may addheat to the circulated fluid by burning a combustible material (e.g.,natural gas). Chiller 22 may place the circulated fluid in a heatexchange relationship with another fluid (e.g., a refrigerant) in a heatexchanger (e.g., an evaporator). The refrigerant removes heat from thecirculated fluid during an evaporation process, thereby cooling thecirculated fluid.

The circulated fluid from chiller 22 or boiler 24 may be transported toAHUs 36 via piping 32. AHUs 36 may place the circulated fluid in a heatexchange relationship with an airflow passing through AHUs 36. Forexample, the airflow may be passed over piping in fan coil units orother air conditioning terminal units through which the circulated fluidflows. AHUs 36 may transfer heat between the airflow and the circulatedfluid to provide heating or cooling for the airflow. The heated orcooled air may be delivered to building 10 via an air distributionsystem including air supply ducts 38 and may return to AHUs 36 via airreturn ducts 40. In FIG. 1, HVAC system 20 is shown to include aseparate AHU 36 on each floor of building 10. In other embodiments, asingle AHU (e.g., a rooftop AHU) may supply air for multiple floors orzones. The circulated fluid from AHUs 36 may return to chiller 22 orboiler 24 via piping 34.

In some embodiments, the refrigerant in chiller 22 is vaporized uponabsorbing heat from the circulated fluid. The vapor refrigerant may beprovided to a compressor within chiller 22 where the temperature andpressure of the refrigerant are increased (e.g., using a rotatingimpeller, a screw compressor, a scroll compressor, a reciprocatingcompressor, a centrifugal compressor, etc.). The compressed refrigerantmay be discharged into a condenser within chiller 22. In someembodiments, water (or another chilled fluid) flows through tubes in thecondenser of chiller 22 to absorb heat from the refrigerant vapor,thereby causing the refrigerant to condense. The water flowing throughtubes in the condenser may be pumped from chiller 22 to a rooftopcooling unit 26 via piping 28. Cooling unit 26 may use fan drivencooling or fan driven evaporation to remove heat from the water. Thecooled water in rooftop unit 26 may be delivered back to chiller 22 viapiping 30 and the cycle repeats.

Referring now to FIG. 2, a block diagram illustrating a portion of HVACsystem 20 in greater detail is shown, according to an exemplaryembodiment. In FIG. 2, AHU 36 is shown as an economizer type airhandling unit. Economizer type air handling units vary the amount ofoutside air and return air used by the air handling unit for heating orcooling. For example, AHU 36 may receive return air 82 from building 10via return air duct 40 and may deliver supply air 86 to building 10 viasupply air duct 38. AHU 36 may be configured to operate exhaust airdamper 60, mixing damper 62, and outside air damper 64 to control anamount of outside air 80 and return air 82 that combine to form supplyair 86. Any return air 82 that does not pass through mixing damper 62may be exhausted from AHU 36 through exhaust damper 60 as exhaust air84.

Each of dampers 60-64 may be operated by an actuator. As shown in FIG.2, exhaust air damper 60 is operated by actuator 54, mixing damper 62 isoperated by actuator 56, and outside air damper 64 is operated byactuator 58. Actuators 54-58 may communicate with an AHU controller 44via a communications link 52. AHU controller 44 may be an economizercontroller configured to use one or more control algorithms (e.g.,state-based algorithms, ESC algorithms, PID control algorithms, modelpredictive control algorithms, etc.) to control actuators 54-58.Exemplary ESC methods that may be used by AHU controller 44 aredescribed in greater detail with reference to FIGS. 6-7.

Actuators 54-58 may receive control signals from AHU controller 44 andmay provide feedback signals to AHU controller 44. Feedback signals mayinclude, for example, an indication of a current actuator or damperposition, an amount of torque or force exerted by the actuator,diagnostic information (e.g., results of diagnostic tests performed byactuators 54-58), status information, commissioning information,configuration settings, calibration data, and/or other types ofinformation or data that may be collected, stored, or used by actuators54-58.

Still referring to FIG. 2, AHU 36 is shown to include a cooling coil 68,a heating coil 70, and a fan 66. In some embodiments, cooling coil 68,heating coil 70, and fan 66 are positioned within supply air duct 38.Fan 66 may be configured to force supply air 86 through cooling coil 68and/or heating coil 70. AHU controller 44 may communicate with fan 66via communications link 78 to control a flow rate of supply air 86.Cooling coil 68 may receive a chilled fluid from chiller 22 via piping32 and may return the chilled fluid to chiller 22 via piping 34. Valve92 may be positioned along piping 32 or piping 34 to control an amountof the chilled fluid provided to cooling coil 68. Heating coil 70 mayreceive a heated fluid from boiler 24 via piping 32 and may return theheated fluid to boiler 24 via piping 34. Valve 94 may be positionedalong piping 32 or piping 34 to control an amount of the heated fluidprovided to heating coil 70.

Each of valves 92-94 may be controlled by an actuator. As shown in FIG.2, valve 92 is controlled by actuator 88 and valve 94 is controlled byactuator 90. Actuators 88-90 may communicate with AHU controller 44 viacommunications links 96-98. Actuators 88-90 may receive control signalsfrom AHU controller 44 and may provide feedback signals to controller44. In some embodiments, AHU controller 44 receives a measurement of thesupply air temperature from a temperature sensor 72 positioned in supplyair duct 38 (e.g., downstream of cooling coil 68 and heating coil 70).However, temperature sensor 72 is not required and may not be includedin some embodiments.

AHU controller 44 may operate valves 92-94 via actuators 88-90 tomodulate an amount of heating or cooling provided to supply air 86(e.g., to achieve a setpoint temperature for supply air 86 or tomaintain the temperature of supply air 86 within a setpoint temperaturerange). The positions of valves 92-94 affect the amount of cooling orheating provided to supply air 86 by cooling coil 68 or heating coil 70and may correlate with the amount of energy consumed to achieve adesired supply air temperature. In various embodiments, valves 92-94 maybe operated by AHU controller 44 or a separate controller for HVACsystem 20.

AHU controller 44 may monitor the positions of valves 92-94 viacommunications links 96-98. AHU controller 44 may use the positions ofvalves 92-94 as the variable to be optimized using an ESC controltechnique. AHU controller 44 may determine and/or set the positions ofdampers 60-64 to achieve an optimal or target position for valves 92-94.The optimal or target position for valves 92-94 may be the position thatcorresponds to the minimum amount of mechanical heating or cooling usedby HVAC system 20 to achieve a setpoint supply air temperature (e.g.,minimum fluid flow through valves 92-94).

Still referring to FIG. 2, HVAC system 20 is shown to include asupervisory controller 42 and a client device 46. Supervisory controller42 may include one or more computer systems (e.g., servers, BAScontrollers, etc.) that serve as enterprise level controllers,application or data servers, head nodes, master controllers, or fieldcontrollers for HVAC system 20. Supervisory controller 42 maycommunicate with multiple downstream building systems or subsystems(e.g., an HVAC system, a security system, etc.) via a communicationslink 50 according to like or disparate protocols (e.g., LON, BACnet,etc.).

In some embodiments, AHU controller 44 receives information (e.g.,commands, setpoints, operating boundaries, etc.) from supervisorycontroller 42. For example, supervisory controller 42 may provide AHUcontroller 44 with a high fan speed limit and a low fan speed limit. Alow limit may avoid frequent component and power taxing fan start-upswhile a high limit may avoid operation near the mechanical or thermallimits of the fan system. In various embodiments, AHU controller 44 andsupervisory controller 42 may be separate (as shown in FIG. 2) orintegrated. In an integrated implementation, AHU controller 44 may be asoftware module configured for execution by a processor of supervisorycontroller 42.

Client device 46 may include one or more human-machine interfaces orclient interfaces (e.g., graphical user interfaces, reportinginterfaces, text-based computer interfaces, client-facing web services,web servers that provide pages to web clients, etc.) for controlling,viewing, or otherwise interacting with HVAC system 20, its subsystems,and/or devices. Client device 46 may be a computer workstation, a clientterminal, a remote or local interface, or any other type of userinterface device. Client device 46 may be a stationary terminal or amobile device. For example, client device 46 may be a desktop computer,a computer server with a user interface, a laptop computer, a tablet, asmartphone, a PDA, or any other type of mobile or non-mobile device.

Extremum-Seeking Control Systems

Referring now to FIG. 3A, a block diagram of an extremum-seeking control(ESC) system 300 is shown, according to an exemplary embodiment. ESCsystem 300 is shown to include an extremum-seeking controller 302 and aplant 304. A plant in control theory is the combination of a process andone or more mechanically-controlled outputs. For example, plant 304 maybe an air handling unit configured to control temperature within abuilding space via one or more mechanically-controlled actuators and/ordampers. In various embodiments, plant 304 may include a chilleroperation process, a damper adjustment process, a mechanical coolingprocess, a ventilation process, a refrigeration process, or any otherprocess where a variable is manipulated to affect an output from plant304.

Extremum-seeking controller 302 uses extremum-seeking control logic tomodulate its output in response to a changing measurement 306 receivedfrom plant 304 via input interface 310. Measurements from plant 304 mayinclude, but are not limited to, information received from sensors aboutthe state of plant 304 or control signals sent to other devices in thesystem. In some embodiments, measurement 306 is a measured or observedposition of one of valves 92-94. In other embodiments, measurement 306is a measured or calculated amount of power consumption, a fan speed, adamper position, a temperature, or any other variable that can bemeasured or calculated by plant 304. Measurement 306 may be the variablethat extremum-seeking controller 302 seeks to optimize (i.e., theperformance variable) via an extremum-seeking control process.Measurement 306 may be output by plant 304 or observed at plant 304(e.g., via a sensor) and provided to extremum-seeking controller atinput interface 310.

Input interface 310 provides measurement 306 to performance gradientprobe 312 to detect the performance gradient. The performance gradientmay indicate a slope of the function y=ƒ(x), where y representsmeasurement 306 and x represents manipulated variable 308. When theperformance gradient is zero, measurement 306 has an extremum value(e.g., a maximum or minimum). Therefore, extremum-seeking controller 302can optimize the value of measurement 306 by driving the performancegradient to zero. Manipulated variable updater 316 produces an updatedmanipulated variable 308 based upon the performance gradient. In anexemplary embodiment, manipulated variable updater 316 includes anintegrator to drive the performance gradient to zero. Manipulatedvariable updater 316 then provides an updated manipulated variable 308to plant 304 via output interface 318. In some embodiments, manipulatedvariable 308 is provided to one of dampers 60-64 (FIG. 2) or an actuatoraffecting dampers 60-64 as a control signal via output interface 318.Plant 304 may use manipulated variable 308 as a setpoint to adjust theposition of dampers 60-64 and thereby control the relative proportionsof outdoor air 80 and recirculation air 82 provided to atemperature-controlled space.

Referring now to FIG. 3B, a block diagram of a dither switchingextremum-seeking control (DSESC) system 350 is shown, according to anexemplary embodiment. DSESC system 350 is shown to include a ditherswitching extremum-seeking controller 352 and plant 304. In DSESC system350, measurement 306 is a function of several manipulated variables 358provided as inputs to plant 304. Manipulated variables 358 may includeany controlled input provided to plant 304 to affect the value ofmeasurement 306. For example, manipulated variables 358 may includesetpoints for multiple chillers, dampers, actuators, or other componentsof HVAC system 20 that can be operated to affect the value ofmeasurement 306 (e.g., a measured supply air temperature, a measuredpower consumption, etc.). Controller 352 includes several of the samecomponents as controller 302 (e.g., performance gradient probe 312,manipulated variable updater 316, input interface 310, and outputinterface 318) and an additional component shown as performance gradientstabilizer 314.

Controller 352 may be configured to perform a dither-demodulationprocess to determine the effect of manipulated variables 358 onmeasurement 306. For example, manipulated variable updater 316 may adddither signals to each of manipulated variables 358. In someembodiments, each of manipulated variables 358 is modified by adifferent dither signal having a different dither frequency. Performancegradient probe 312 may pass measurement 306 through multiple parallelhigh-pass filters (e.g., one for each dither signal) and apply ademodulation signal to the output of each high-pass filter. Thefrequency and phase of each demodulation signal may be selected tomaximize the cross-correlation between the demodulation signal and theoutput of the corresponding high-pass filter. Performance gradient probe312 may multiply the output of each high-pass filter by a differentdemodulation signal and provide each product to a low-pass filter. Thecutoff frequencies of the high-pass and low-pass filters may be selectedto extract gradient information from measurement 306. An exemplarydither-demodulation process which may be performed by controller 352 isdescribed in greater detail with reference to FIG. 4.

In DSESC system 350, the outputs from performance gradient probe 312 areprovided to a performance gradient stabilizer 314. Performance gradientstabilizer 314 may include, for example, hysteresis devices and/orflip/flop devices configured to stabilize the performance gradientswhile rejecting the negative impact of measurement noise. Performancegradient stabilizer 314 provides the stabilized performance gradients tomanipulated variable updater 316, which uses integrators to drive theperformance gradients to zero. In some embodiments, DSESC system 350 isthe same or similar to the DSESC system described in U.S. patentapplication Ser. No. 14/538,700, filed Nov. 11, 2014, the entiredisclosure of which is incorporated by reference herein.

Referring now to FIG. 4, a block diagram of another extremum-seekingcontrol (ESC) system 400 is shown, according to an exemplary embodiment.ESC system 400 is shown to include a plant 404 and an extremum-seekingcontroller 402. Controller 402 uses an extremum-seeking control strategyto optimize a performance variable y received as an output from plant404. Performance variable y may be the same or similar to measurement306, as described with reference to FIGS. 3A-3B. For example,performance variable y may be the variable that ESC controller 402 seeksto optimize using an extremum-seeking control process. Optimizingperformance variable y may include minimizing y, maximizing y,controlling y to achieve a setpoint, or otherwise regulating the valueof performance variable y.

Plant 404 may be the same or similar to plant 304, as described withreference to FIGS. 3A-3B. For example, plant 404 may be a combination ofa process and one or more mechanically-controlled outputs. In someembodiments, plant 404 is an air handling unit configured to controltemperature within a building space via one or moremechanically-controlled actuators and/or dampers. In other embodiments,plant 404 may include a chiller operation process, a damper adjustmentprocess, a mechanical cooling process, a ventilation process, or anyother process that generates an output based on one or more controlinputs.

Plant 404 may be represented mathematically as a combination of inputdynamics 406, a performance map 408, output dynamics 410, and noise n.In some embodiments, input dynamics 406 are linear time-invariant (LTI)input dynamics and output dynamics 410 are LTI output dynamics.Performance map 408 may be a static nonlinear performance map. Noise nmay be process noise, measurement noise, or a combination of both. Theactual mathematical model for plant 404 does not need to be known inorder to apply ESC and is illustrative only.

Plant 404 receives a control input u (e.g., a control signal, amanipulated variable, etc.) from output interface 416 ofextremum-seeking controller 402. Input dynamics 406 may use the controlinput u to generate a function signal x based on the control input(e.g., x=ƒ(u)). Function signal x may be passed to performance map 408which generates an output signal z as a function of the function signal(i.e., z=ƒ(x)). Extremum-seeking controller 402 may seek to find valuesfor x and/or u that optimize the output z of performance map 408. Theoutput signal z may be passed through output dynamics 410 to producesignal z′, which is modified by noise n to produce performance variabley (e.g., y=z′+n). Performance variable y is provided as an output fromplant 404 and received at input interface 414 of extremum-seekingcontroller 402.

Still referring to FIG. 4, extremum-seeking controller 402 is shownreceiving performance variable y at input interface 414 and providingperformance variable y to a control loop 405 within controller 402.Control loop 405 is shown to include a high-pass filter 418, ademodulation element 428, a low-pass filter 420, an integrator 422, anda dither signal element 430. Control loop 405 may be configured toproduce a performance gradient p from performance variable y using adither-demodulation process. Integrator 422 analyzes the performancegradient p and adjusts the value of signal v to drive performancegradient p to zero.

The first step of the dither-demodulation process is performed by dithersignal generator 426 and dither signal element 430. Dither signalelement 430 receives a dither signal d from dither signal generator 426and the signal v from integrator 422. Dither signal element 430 combinesdither signal d with the signal v from integrator 422 to perturb thecontrol input u provided to plant 404 (e.g., u=v+d). The perturbedcontrol input u is provided to plant 404 via output interface 416 andused by plant 404 to generate performance variable y as previouslydescribed.

The second step of the dither-demodulation process is performed byhigh-pass filter 418, demodulation element 428, and low-pass filter 420.High-pass filter 418 filters the performance variable y and provides thefiltered output to demodulation element 428. Demodulation element 428demodulates the output of high-pass filter 418 by multiplying thefiltered output by a demodulation signal m generated by a demodulationsignal generator 424. Demodulation element 428 provides the demodulatedoutput to low-pass filter 420, which extracts a performance gradient pfrom the demodulated output. The cutoff frequencies of high-pass filter418 and low-pass filter 420 may be selected to extract gradientinformation from performance variable y at a particular ditherfrequency. In some embodiments, the demodulation signal m has the samefrequency as the dither signal d to enhance the extraction ofperformance gradient p with respect to dither signal d.

Self-Configuring Extremum-Seeking Control System

Referring now to FIG. 5, a block diagram of a self-configuringextremum-seeking control (SCESC) system 500 is shown, according to anexemplary embodiment. SCESC system 500 is shown to include a plant 503and a self-configuring extremum-seeking controller 501. Controller 501may receive feedback from plant 503 via input interface 534 and providecontrol inputs to plant 503 via output interface 536. Controller 501 mayoperate in a manner similar to controllers 302, 352, and 402, asdescribed with reference to FIGS. 3A-4. For example, controller 501 mayuse an extremum-seeking control (ESC) strategy to optimize a performancevariable y received as an output from plant 503. Controller 501 mayperturb the control input u to plant 503 with a dither signal d andanalyze the effect of such perturbation on performance variable y.Controller 501 may adjust the control input u to drive the gradient ofperformance variable y to zero. In this way, controller 501 identifiesvalues for control input u that achieve an optimal value (e.g., amaximum or a minimum) for performance variable y.

In some embodiments, the ESC logic implemented by controller 501generates values for control input u based on a received control signal(e.g., a setpoint, an operating mode signal, etc.). The control signalmay be received from a user control (e.g., a thermostat, a local userinterface, etc.), client devices 514 (e.g., computer terminals, mobileuser devices, cellular phones, laptops, tablets, desktop computers,etc.), a supervisory controller 512, or any other external system ordevice. In various embodiments, controller 501 may communicate withexternal systems and devices directly (e.g., using NFC, Bluetooth, WiFidirect, cables, etc.) or via a communications network 510 (e.g., aBACnet network, a LonWorks network, a LAN, a WAN, the Internet, acellular network, etc.) using wired or wireless electronic datacommunications.

Although ESC does not require a model of the system to optimizeperformance variable y, properly configuring an ESC system may requireknowledge of the system bandwidth (e.g., the natural frequency ω_(n) ofplant 503) to select the frequency ω_(d) of the dither signal d. Forexample, it may be desirable to select a dither signal frequency ω_(d)that is the same or similar to the natural frequency ω_(n) of plant 503to enhance the effect of dither signal d on the performance variable y(e.g., by increasing the resonance between dither signal d and plant503). Additionally, it may be desirable to know the system gain K_(s) inorder to properly set the rate of gradient descent. Advantageously,SCESC controller 501 may be configured to automatically determine boththe system bandwidth and the system gain using the dither signal d toperform system identification.

In some embodiments, SCESC controller 501 determines the naturalfrequency ω_(n) of plant 503 and uses the natural frequency ω_(n) as thesystem bandwidth. SCESC controller 501 may determine the naturalfrequency ω_(n) based on the phase delay φ between the dither signal dand the performance variable y. For example, SCESC controller 501 mayexcite plant 503 with a dither signal d having a known dither frequencyω_(d) and may measure the resultant phase delay φ of the performancevariable y relative to the dither signal d. SCESC controller 501 may usethe phase delay φ in conjunction with the dither signal frequency ω_(d)to estimate the natural frequency ω_(n), as shown in the followingequation:

$\omega_{n} = {- {\omega_{d}\left( {{\cot (\varphi)} - \sqrt{\frac{1}{\sin^{2}(\varphi)}}} \right)}}$

SCESC controller 501 may use the natural frequency ω_(n) to set orupdate the dither signal frequency ω_(d) (e.g., by setting the dithersignal frequency ω_(d) equal to the natural frequency ω_(n)).

In some embodiments, SCESC controller 501 uses a history of values forthe dither signal d to calculate an exponentially-weighted movingaverage (EWMA) s_(d,k) of the dither signal d at time k. Similarly,SCESC controller 501 may use a history of values for the performancevariable y to calculate an EWMA s_(y,k) of the performance variable y attime k. SCESC controller 501 may use the EWMAs s_(d,k) and s_(y,k) inconjunction with the natural frequency ω_(n) and the known dither signalfrequency ω_(d) to estimate the system gain K_(s), as shown in thefollowing equation:

$K_{s} = \sqrt{\frac{s_{y,k}}{s_{d,k}}\left( {\left( {1 - \left( \frac{\omega_{d}}{\omega_{n}} \right)^{2}} \right)^{2} + \left( {2\zeta \frac{\omega_{d}}{\omega_{n}}} \right)^{2}} \right)}$

SCESC controller 501 may use the system gain K_(s) to set or update therate of gradient descent. Advantageously, automatically determining thesystem bandwidth and the system gain enables SCESC system 500 to beconfigured automatically (e.g., without manual testing andconfiguration) and updated periodically as the ESC parameters and/orsystem conditions change.

Still referring to FIG. 5, controller 501 is shown to include acommunications interface 502, an input interface 534, and an outputinterface 536. Interfaces 502 and 534-536 may include any number ofjacks, wire terminals, wire ports, wireless antennas, or othercommunications interfaces for communicating information and/or controlsignals. Interfaces 502 and 534-536 may be the same type of devices ordifferent types of devices. For example, input interface 534 may beconfigured to receive an analog feedback signal (e.g., an outputvariable, a measured signal, a sensor output, a controlled variable)from plant 503, whereas communications interface 502 may be configuredto receive a digital setpoint signal from upstream supervisorycontroller 512 via network 510. Output interface 536 may be a digitaloutput (e.g., an optical digital interface) configured to provide adigital control signal (e.g., a manipulated variable, a control input)to plant 503. In other embodiments, output interface 536 is configuredto provide an analog output signal.

In some embodiments interfaces 502 and 534-536 can be joined as one ortwo interfaces rather than three separate interfaces. For example,communications interface 502 and input interface 534 may be combined asone Ethernet interface configured to receive network communications fromsupervisory controller 512. In some embodiments, supervisory controller512 provides both a setpoint and process feedback via an Ethernetnetwork (e.g., network 510). In such an embodiment, output interface 536may be specialized for a controlled process component of plant 503. Inother embodiments, output interface 536 can be another standardizedcommunications interface for communicating data or control signals.Interfaces 502 and 534-536 can include communications electronics (e.g.,receivers, transmitters, transceivers, modulators, demodulators,filters, communications processors, communication logic modules,buffers, decoders, encoders, encryptors, amplifiers, etc.) configured toprovide or facilitate the communication of the signals described herein.

Still referring to FIG. 5, controller 501 is shown to include aprocessing circuit 504 having a processor 506 and memory 508. Processor506 may be a general purpose or specific purpose processor, anapplication specific integrated circuit (ASIC), one or more fieldprogrammable gate arrays (FPGAs), a group of processing components, orother suitable processing components. Processor 506 is configured toexecute computer code or instructions stored in memory 508 or receivedfrom other computer readable media (e.g., CDROM, network storage, aremote server, etc.).

Memory 508 may include one or more devices (e.g., memory units, memorydevices, storage devices, etc.) for storing data and/or computer codefor completing and/or facilitating the various processes described inthe present disclosure. Memory 508 may include random access memory(RAM), read-only memory (ROM), hard drive storage, temporary storage,non-volatile memory, flash memory, optical memory, or any other suitablememory for storing software objects and/or computer instructions. Memory508 may include database components, object code components, scriptcomponents, or any other type of information structure for supportingthe various activities and information structures described in thepresent disclosure. Memory 508 may be communicably connected toprocessor 506 via processing circuit 504 and may include computer codefor executing (e.g., by processor 506) one or more processes describedherein.

Still referring to FIG. 5, memory 508 is shown to include a signalfilter 516. Signal filter 516 may be configured to filter the incomingperformance variable signal y and the dither signal d to remove DCvalues from the two signals. In some embodiments, the performancevariable signal y and the dither signal d have DC values that changeslowly over time. Signal filter 516 may include a high-pass filter, afirst-difference filter, or any other type of filter that filters theperformance variable signal y and the dither signal d to remove the DCvalues from the signals. In other embodiments, the performance variablesignal y and the dither signal d have constant DC values and signalfilter 516 may be omitted.

In some embodiments, signal filter 516 smoothes the performance variablesignal y and the dither signal d to reduce the effect of noise on suchsignals. For example, signal filter 516 may include a Savitzky Golayfilter which can be applied to the differences of the performancevariable signal y and the dither signal d to reduce the impact of noisewithout significantly affecting the phase of such signals. Signal filter516 may provide a filtered performance variable signal y_(ƒ) and afiltered dither signal d_(ƒ) to EWMA calculator 518.

In some embodiments, signal filter 516 includes the functionality ofhigh-pass filter 418, demodulation element 428, and/or low-pass filter420, as described with reference to FIG. 4. For example, signal filter516 may be configured to receive the feedback signal y from plant 503via input interface 534 and to filter the feedback signal y using one ormore high-pass filters and/or low-pass filters. The number of high-passfilters and low-pass filters used by signal filter 516 may depend on thenumber N of control inputs u provided to plant 503. In some embodiments,signal filter 516 includes a high-pass filter and a low-pass filter foreach control input u. The cutoff frequencies of each high-pass filterand each low-pass filter may be selected to facilitate extractingperformance gradient information from the feedback signal y at aparticular dither frequency. In some embodiments, the values of thecutoff frequencies are determined by dither signal generator 530 basedon the dither signal frequency ω_(d). Signal filter 516 may retrievesuch values from dither signal generator 530 and use the values to tunethe high-pass and low-pass filters.

In some embodiments, signal filter 516 includes a demodulationcomponent. For example, signal filter 516 may be configured to generateone or more demodulation signals for use in demodulating the outputsfrom the high-pass filters. The number of demodulation signals generatedmay depend on the number of control inputs u (e.g., a separatedemodulation signal for each control input u) and/or the number ofhigh-pass filters (e.g., a separate demodulation signal for eachhigh-pass filter). In some embodiments, each demodulation signal has aphase shift selected to maximize the cross-correlation between thedemodulation signal and the output of the corresponding high-passfilter. In some embodiments, the demodulation frequencies and the phaseshifts for the demodulation signals are determined by dither signalgenerator 530. Signal filter 516 may retrieve such values from dithersignal generator 530 and use the values to generate the demodulationsignals.

Signal filter 516 may include one or more demodulation elements (e.g.,demodulation element 428) configured to multiply each demodulationsignal with the output of the corresponding high-pass filter. Signalfilter 516 may provide the product of each multiplication as an input tothe corresponding low-pass filter. Advantageously, using multipledifferent dither signals and demodulation signals allows each controlinput to be independently isolated by defining different ditherfrequencies for each dither/demodulation signal pair. Accordingly,controller 501 can attribute a change in the performance variable signaly to a change in a particular control input u. Changes in theperformance variable signal y caused by each control input u can bedistinguished from each other and distinguished from changes caused byexternal disturbances and/or process noise.

Still referring to FIG. 5, memory 508 is shown to include anexponentially-weighted moving average (EWMA) calculator 518. EWMAcalculator 518 may be configured to calculate EWMAs based on a historyof time-series values of the performance variable signal y and thedither signal d. In some embodiments, EWMA calculator 518 uses theoriginal performance variable signal y and the original dither signal dto calculate the EWMAs. In other embodiments, EWMA calculator 518 usesthe filtered performance variable signal y_(ƒ) and the filtered dithersignal d_(ƒ) to calculate the EWMAs. EWMA calculator 518 may recursivelyupdate the EWMA values each time a new value for the performancevariable signal y_(ƒ) and the dither signal d_(ƒ) is received.Advantageously, EWMA calculator 518 allows controller 501 to track theperformance variable signal y_(ƒ) and the dither signal d_(ƒ) withoutstoring a batch of previous data values for each signal.

EWMA calculator 518 may generate a first EWMA s_(d,k) based on a historyof values for the dither signal d or the filtered dither signal d_(ƒ).The notation s_(d,k) indicates that the variable s_(d,k) represents theEWMA of the dither signal d (or filtered dither signal d_(ƒ)) at time k.EWMA calculator 518 may generate the first EWMA s_(d,k) using thefollowing equation:

$s_{d,k} = {s_{d,{k - 1}} + \frac{d_{k}^{2} - s_{d,{k - 1}}}{\min \left( {k,N_{p}} \right)}}$

where s_(d,k) is the EWMA value at time k, s_(d,k-1) is the previousEWMA value (i.e., at time k−1), d_(k) is the value of the dither signald or filtered dither signal d_(ƒ) at time k, and N_(p) is the durationof the forgetting window. In some embodiments, EWMA calculator 518generates a value for N_(p) based on the dither signal frequency ω_(d)as shown in the following equation:

$N_{p} = \frac{10\left( {2\pi} \right)}{\omega_{d}}$

EWMA calculator 518 may generate a second EWMA s_(y,k) based on ahistory of values for the performance variable signal y or the filteredperformance variable signal y_(ƒ). The notation s_(y,k) indicates thatthe variable s_(y,k) represents the EWMA of the performance variablesignal y (or the filtered performance variable signal y_(ƒ)) at time k.EWMA calculator 518 may generate the second EWMA s_(y,k) using thefollowing equation:

$s_{y,k} = {s_{y,{k - 1}} + \frac{y_{k}^{2} - s_{y,{k - 1}}}{\min \left( {k,N_{p}} \right)}}$

where s_(y,k) is the EWMA value at time k, s_(y,k-1) is the previousEWMA value (i.e., at time k−1), y_(k) is the value of the performancevariable signal y or the filtered performance variable signal y_(ƒ) attime k, and N_(p) is the duration of the forgetting window.

In some embodiments, EWMA calculator 518 generates a third EWMA s_(x,k)based on a product of the performance variable signal y (or the filteredperformance variable signal y_(ƒ)) and the dither signal d (or thefiltered dither signal d_(ƒ)). EWMA calculator 518 may generate thethird EWMA s_(x,k) using the following equation:

$s_{x,k} = {s_{x,{k - 1}} + \frac{{d_{k}y_{k}} - s_{x,{k - 1}}}{\min \left( {k,N_{p}} \right)}}$

where s_(x,k) is the EWMA value at time k, s_(x,k-1) is the previousEWMA value (i.e., at time k−1), d_(k) is the value of the dither signald or filtered dither signal d_(ƒ) at time k, y_(k) is the value of theperformance variable signal y or the filtered performance variablesignal y_(ƒ) at time k, and N_(p) is the duration of the forgettingwindow.

In some embodiments, EWMA calculator 518 updates the EWMA valuess_(d,k), s_(y,k), and s_(x,k) each time a new sample of the performancevariable y and the dither signal d is received (e.g., at the beginningof each time step k). EWMA calculator 518 may provide the EWMA valuess_(d,k), s_(y,k), and s_(x,k) to phase delay estimator 520 for use incalculating the phase delay φ between the dither signal d and theperformance variable signal y.

Still referring to FIG. 5, memory 508 is shown to include a phase delayestimator 520. Phase delay estimator 520 may be configured to estimatethe phase delay φ between the dither signal d and the performancevariable signal y. In some embodiments, phase delay estimator 520estimates the phase delay φ using the original or filtered values forthe performance variable signal y and the dither signal d. In otherembodiments, phase delay estimator 520 estimates the phase delay φ usingthe EWMA values s_(d,k), s_(y,k), and s_(x,k). Both embodiments aredescribed in greater detail below.

According to the first exemplary embodiment, phase delay estimator 520estimates the phase delay φ using the original or filtered values forthe performance variable signal y and the dither signal d. For example,phase delay estimator 520 may estimate the phase delay φ using thefollowing property of vectors:

$\varphi = {\cos^{- 1}\left( \frac{u_{1} \cdot u_{2}}{{u_{1}}_{2}{u_{2}}_{2}} \right)}$

where u₁ and u₂ are equal length vectors, u₁·u₂ is the dot product ofthe two vectors, and the notation ∥ ∥₂ refers to the 2-norm. The dotproduct u₁·u₂ can be calculated using the following equation:

${u_{1} \cdot u_{2}} = {\sum\limits_{k = 1}^{N}\; {u_{1,k}u_{2,k}}}$

and the 2-norms can be calculated using the following equations:

${u_{1}}_{2} = {{\sqrt{\sum\limits_{k = 1}^{N}\; u_{1,k}^{2}}\mspace{31mu} {u_{2}}_{2}} = \sqrt{\sum\limits_{k = 1}^{N}\; u_{2,k}^{2}}}$

Phase delay estimator 520 may substitute the dither signal d (or thefiltered dither signal d_(ƒ)) for u₁ and the performance variable signaly (or the filtered performance variable signal y_(ƒ)) for u₂ to estimatethe phase delay between the dither signal and the performance variablesignal.

According to the second exemplary embodiment, phase delay estimator 520estimates the phase delay φ using the EWMA values s_(d,k), s_(y,k), ands_(x,k). For example, phase delay estimator 520 may estimate the phasedelay {circumflex over (φ)}_(k) at time k using the following equation:

${\hat{\varphi}}_{k} = {\cos^{- 1}\left( \frac{s_{x,k}}{\sqrt{s_{d,k}s_{y,k}}} \right)}$

Phase delay estimator 520 may provide the phase delay φ and/or{circumflex over (φ)}_(k) to bandwidth estimator 524 for use inestimating the system bandwidth.

Still referring to FIG. 5, memory 508 is shown to include a bandwidthestimator 524. Bandwidth estimator 524 may estimate the naturalfrequency ω_(n) of the system and use the natural frequency ω_(n) as thesystem bandwidth. Bandwidth estimator 524 may estimate the naturalfrequency ω_(n) based on the assumption that the transfer functionbetween the dither signal d and the performance variable signal y can beapproximated as second order, as shown in the following equation:

${G(s)} \approx \frac{\omega_{n}^{2}}{s^{2} + {2{\zeta\omega}_{n}s} + \omega_{n}^{2}}$

where ζ is the damping coefficient and ω_(n) is the natural frequency ofthe system. The natural frequency ω_(n) may be assumed to be equivalentto the system bandwidth.

From this transfer function, it follows that the phase delay φ betweenthe dither signal d and the performance variable signal y is:

$\varphi = {- {\tan^{- 1}\left( \frac{2\zeta \frac{\omega_{d}}{\omega_{n}}}{1 - \frac{\omega_{d}^{2}}{\omega_{n}^{2}}} \right)}}$

which can be simplified by assuming that plant 503 is critically damped(i.e., ζ=1). The previous equation can be rearranged into a quadraticform that provides ω_(n) as a function of the dither signal frequencyω_(d) and the phase delay φ as follows:

${\omega_{n}^{2} + {\frac{2\omega_{d}}{\tan (\varphi)}\omega_{n}} - \omega_{d}^{2}} = 0$

This quadratic equation can be solved to give two frequencies for ω_(n),one positive frequency and one negative frequency. The positivefrequency is given by:

$\omega_{n} = {- {\omega_{d}\left( {{\cot (\varphi)} - \sqrt{\frac{1}{\sin^{2}(\varphi)}}} \right)}}$

For the embodiment in which phase delay estimator 520 uses the originalor filtered values of the dither signal d and the performance variablesignal y to estimate the phase delay φ, bandwidth estimator 524 mayestimate the natural frequency ω_(n) as a function of the known dithersignal frequency ω_(d) and the phase delay φ using the previousequation. For example, bandwidth estimator 524 is shown receiving thephase delay φ from phase delay estimator 520 and the dither signalfrequency ω_(d) from dither signal generator 530. Bandwidth estimator524 may use the dither signal frequency ω_(d) and the phase delay φ tocalculate ω_(n) as shown in the previous equation. Bandwidth estimator524 may store or use the natural frequency ω_(n) as the systembandwidth.

For the embodiment in which phase delay estimator 520 uses the EWMAvalues s_(d,k), s_(y,k), and s_(x,k) to estimate the phase delay{circumflex over (φ)}_(k), bandwidth estimator 524 may estimate thenatural frequency {circumflex over (ω)}_(n,k) at time k using thefollowing equation:

${\hat{\omega}}_{n,k} = {- {\omega_{d}\left( {{\cot \left( {\hat{\varphi}}_{k} \right)} - \sqrt{\frac{1}{\sin^{2}\left( {\hat{\varphi}}_{k} \right)}}} \right)}}$

For example, bandwidth estimator 524 may receive the estimated phasedelay {circumflex over (φ)}_(k) from phase delay estimator 520 and thedither signal frequency ω_(d) from dither signal generator 530.Bandwidth estimator 524 may use the dither signal frequency ω_(d) andthe estimated phase delay {circumflex over (φ)}_(k) to calculate{circumflex over (ω)}_(n,k) as shown in the previous equation. Bandwidthestimator 524 may store or use the natural frequency ω_(n,k) as thesystem bandwidth at time k.

Still referring to FIG. 5, memory 508 is shown to include a gainestimator 526. Gain estimator 526 may be configured to estimate thesystem gain K_(s) based on the natural frequency ω_(n) or {circumflexover (ω)}_(n,k) received from bandwidth estimator 524. The system gainK_(s) may be used by controller 501 (e.g., integrator 528) to scale thestep size of the internal gradient descent procedure employed bycontroller 501.

In some embodiments, gain estimator 526 estimates the system gain K_(s)based on the following equation describing the magnitude of asecond-order system:

${{H\left( {j\; \omega_{d}} \right)}} = \frac{K_{s}}{\sqrt{\left( {1 - \left( \frac{\omega_{d}}{\omega_{n}} \right)^{2}} \right)^{2} + \left( {2\zeta \frac{\omega_{d}}{\omega_{n}}} \right)^{2}}}$

where the system gain K_(s) is unknown. The actual magnitude of theperformance variable signal y relative to the dither signal d is theratio of the ranges of the performance variable signal y and the dithersignal d. In other words, the magnitude |H(jω_(d))| may be equal to theratio of the range of the performance variable signal y (i.e., σ_(y)) tothe range of the dither signal d (i.e., σ_(d)), as shown in thefollowing equation:

${{H\left( {j\; \omega_{d}} \right)}} = \frac{\sigma_{y}}{\sigma_{d}}$

In various embodiments, the ranges σ_(y) and σ_(d) may represent adifference between a maximum and minimum value of the correspondingsignal, a standard deviation of the signal, or any other metric thatquantifies a range or spread of a plurality of data values.

In some embodiments, gain estimator 526 uses the EWMA values s_(d,k) ands_(y,k) as the variances of the dither signal d and the performancevariable signal y, respectively. These EWMA values are measures of thevariable ranges squared (e.g., s_(d,k)=σ_(d) ², and s_(y,k)=σ_(y) ²).Accordingly, the square root of the ratio

$\frac{s_{y,k}}{s_{d,k}}$

may be substituted for |H(jω_(d))| as shown in the following equation:

$\sqrt{\frac{s_{y,k}}{s_{d,k}}} = \frac{K_{s}}{\sqrt{\left( {1 - \left( \frac{\omega_{d}}{\omega_{n}} \right)^{2}} \right)^{2} + \left( {2\zeta \frac{\omega_{d}}{\omega_{n}}} \right)^{2}}}$

which can be solved for K_(s) as follows:

$K_{s} = \sqrt{\frac{s_{y,k}}{s_{d,k}}\left( {\left( {1 - \left( \frac{\omega_{d}}{\omega_{n}} \right)^{2}} \right)^{2} + \left( {2\zeta \frac{\omega_{d}}{\omega_{n}}} \right)^{2}} \right)}$

Gain estimator 526 is shown receiving the EWMA values s_(d,k) ands_(y,k) from EWMA calculator 518, the natural frequency ω_(n) frombandwidth estimator 524, and the dither signal frequency ω_(d) fromdither signal generator 530. Gain estimator 526 may use the previousequation to calculate K_(s) as a function of the EWMA values s_(d,k) ands_(y,k), the natural frequency ω_(n) and the dither signal frequencyω_(d), assuming that the plant is critically damped (i.e., ζ=1). In someembodiments, gain estimator 526 provides the system gain K_(s) tointegrator 528 for use in scaling the step size of the internal gradientdescent procedure performed by controller 501.

Still referring to FIG. 5, memory 508 is shown to include an integrator528. Integrator 528 may be the same or similar to integrator 422, asdescribed with reference to FIG. 4. For example, integrator 528 mayreceive the performance gradient p from signal filter 516 and mayoperate to drive the performance gradient p to zero. For embodiments inwhich controller 501 generates multiple control inputs u, integrator 528includes a plurality of integrators (e.g., one for each control inputu). Integrator 528 may use the system gain K_(s) as an integral gain(e.g., K_(s)/s) to generate a value for the manipulated variable v. Forexample, integrator 528 may generate manipulated variable values v thatdrive the performance gradient p to zero. The variable v may besubsequently modified by feedback controller 532 to generate the controlinput u (e.g., by modifying the variable v with the dither signal d).

Still referring to FIG. 5, memory 508 is shown to include a dithersignal generator 530. Dither signal generator 530 may be configured togenerate one or more dither signals for use in perturbing the controlinputs u provided to plant 503. The number of dither signals generatedby dither signal generator 530 may depend on the number of controlinputs u. In some embodiments, dither signal generator 530 generates adither signal d for each of the control inputs u. Dither signalgenerator 530 may generate the dither signal d according to signalparameters such as a dither amplitude A_(d) and a dither frequencyω_(d). In other words, the dither signal d may have a dither amplitudeA_(d) and a dither frequency ω_(d).

Dither signal generator 530 may select the dither amplitude A_(d) suchthat the input perturbation is no larger than 10% of the total inputrange. In some embodiments, dither signal generator 530 selects thedither amplitude A_(d) such that the corresponding dithered outputamplitude of performance variable signal y (i.e., the amplitude ofperturbation caused by dither signal d) is at least twice the noiseamplitude. Dither signal generator 530 may select the dither frequencyω_(d) based on the natural frequency ω_(n) or natural frequency estimate{circumflex over (ω)}_(n,k) received from bandwidth estimator 524. Forexample, dither signal generator 530 may select the dither frequencyω_(d) to be equal to the natural frequency ω_(n) or a multiple of thenatural frequency ω_(n).

Dither signal generator 530 may provide the dither frequency ω_(d) tobandwidth estimator 524 for use in calculating the natural frequencyω_(n). Dither signal generator 530 may also provide the dither frequencyω_(d) to gain estimator 526 for use in calculating the system gainK_(s). Dither signal generator 530 may provide the dither signal d tofeedback controller 532 for use in generating the control input u.

Still referring to FIG. 5, memory 508 is shown to include a feedbackcontroller 532. Feedback controller 532 may be configured to operatecontroller 501 in a closed-loop operating mode. Feedback controller 532may include one or more processing elements (e.g., dither signal element430) configured to generate a perturbed control input u using thegenerated dither signal d. For example, feedback controller 532 maygenerate a perturbed control input u by adding the dither signal d tothe output v provided by integrator 528 (e.g., u=v+d). In someembodiments, feedback controller 532 receives multiple outputs v fromintegrator 528 and multiple dither signals d from dither signalgenerator 530. Feedback controller 532 may modify each output v using adifferent dither signal d having a different dither frequency. In otherembodiments, feedback controller 532 receives a single dither signal dand a single output v as shown in FIG. 5. Feedback controller 532 mayprovide the perturbed control input u to plant 503 via output interface536.

Self-Configuring Extremum-Seeking Control Processes

Referring now to FIG. 6, a flowchart of a process 600 for automaticallyconfiguring an extremum-seeking control (ESC) system is shown, accordingto an exemplary embodiment. Process 600 may be performed by one or morecomponents of self-configuring extremum-seeking controller 501 toautomatically determine ESC parameters such as the dither signalfrequency ω_(d) and the system gain K_(s). Controller 501 may use theESC parameters to operate a plant (e.g., plant 503) using anextremum-seeking control technique.

Process 600 is shown to include receiving an initial value for a ditherfrequency ω_(d) (step 602). The initial value for the dither frequencyω_(d) may be retrieved from memory, specified by a user, received froman external system or device, automatically determined by controller501, or obtained from any other data source. In some embodiments, theinitial value of the dither frequency ω_(d) is based on knowledge of thesystem. For example, the initial value of the dither frequency ω_(d) maybe determined using a system identification procedure. In otherembodiments, the initial value of the dither frequency ω_(d) is astandard value pre-programmed into controller 501. Advantageously, theinitial value of the dither frequency ω_(d) does not need to be selectedbased on knowledge of the system since the dither frequency ω_(d) isautomatically updated in subsequent steps of process 600.

Process 600 is shown to include updating extremum-seeking control (ESC)parameters using the dither frequency ω_(d) (step 604). ESC controlparameters may include any parameters or variables used by controller501 when performing an extremum-seeking control process. For example,step 604 may include generating a dither signal d using the ditherfrequency ω_(d). In some embodiments, step 604 includes receiving anoutput signal y from the plant, which may be the same as the performancevariable signal y previously described. Throughout this disclosure theterms “performance variable signal y” and “output signal y” are usedinterchangeably to refer to the monitored variable y received from theplant. ESC control parameters generated in step 604 may include a dotproduct of the dither signal d and the output signal y and/or 2-norms ofthe dither signal d and the output signal y. For example, step 604 mayinclude calculating the dot product of the dither signal d and theoutput signal y using the following equation:

${u_{1} \cdot u_{2}} = {\sum\limits_{k = 1}^{N}\; {u_{1,k}u_{2,k}}}$

where u₁ is the dither signal and u₂ is the output signal y. Step 604may include calculating 2-norms of dither signal d and the output signaly using the following equations:

${u_{1}}_{2} = {{\sqrt{\sum\limits_{k = 1}^{N}\; u_{1,k}^{2}}\mspace{31mu} {u_{2}}_{2}} = \sqrt{\sum\limits_{k = 1}^{N}\; u_{2,k}^{2}}}$

where u₁ is the dither signal and u₂ is the output signal y.

In some embodiments, step 604 includes calculating EWMAs of the dithersignal d and the output signal y. For example, step 604 may includegenerating a first EWMA s_(d,k) based on a history of values for thedither signal d or the filtered dither signal d_(ƒ). The notations_(d,k) indicates that the variable s_(d,k) represents the EWMA of thedither signal d (or filtered dither signal d_(ƒ)) at time k. The firstEWMA s_(d,k) may be generated using the following equation:

$s_{d,k} = {s_{d,{k - 1}} + \frac{d_{k}^{2} - s_{d,{k - 1}}}{\min \left( {k,N_{p}} \right)}}$

where s_(d,k) is the EWMA value at time k, s_(d,k-1) is the previousEWMA value (i.e., at time k−1), d_(k) is the value of the dither signald or filtered dither signal d_(ƒ) at time k, and N_(p) is the durationof the forgetting window. In some embodiments, step 604 includesgenerating a value for N_(p) based on the dither signal frequency ω_(d)as shown in the following equation:

$N_{p} = \frac{10\left( {2\pi} \right)}{\omega_{d}}$

Step 604 may include generating a second EWMA s_(y,k) based on a historyof values for the output signal y or the filtered output signal y_(ƒ).The notation s_(y,k) indicates that the variable s_(y,k) represents theEWMA of the output signal y (or the filtered output signal y_(ƒ)) attime k. The second EWMA s_(y,k) may be generated using the followingequation:

$s_{y,k} = {s_{y,{k - 1}} + \frac{y_{k}^{2} - s_{y,{k - 1}}}{\min \left( {k,N_{p}} \right)}}$

where s_(y,k) is the EWMA value at time k, s_(y,k-1) is the previousEWMA value (i.e., at time k−1), y_(k) is the value of the output signaly or the filtered output signal y_(ƒ) at time k, and N_(p) is theduration of the forgetting window.

In some embodiments, step 604 includes generating a third EWMA s_(x,k)based on a product of the output signal y (or the filtered output signaly_(ƒ)) and the dither signal d (or the filtered dither signal d_(ƒ)).The third EWMA s_(x,k) may be generated using the following equation:

$s_{x,k} = {s_{x,{k - 1}} + \frac{{d_{k}y_{k}} - s_{x,{k - 1}}}{\min \left( {k,N_{p}} \right)}}$

where s_(x,k) is the EWMA value at time k, s_(x,k-1) is the previousEWMA value (i.e., at time k−1), d_(k) is the value of the dither signald or filtered dither signal d_(ƒ) at time k, y_(k) is the value of theoutput signal y or the filtered output signal y_(ƒ) at time k, and N_(p)is the duration of the forgetting window. In some embodiments, step 604includes updating the EWMA values s_(d,k), s_(y,k), and s_(x,k) eachtime a new sample of the output signal y and the dither signal d isreceived (e.g., at the beginning of each time step k).

Step 604 may include calculating a value for the system gain K_(s). Thesystem gain K_(s) may be calculated at the beginning of each time step kbased on the current EWMA values s_(d,k) and s_(y,k), the current ditherfrequency ω_(d), and the current estimate of the system bandwidth (e.g.,natural frequency ω_(n)) as shown in the following equation:

$K_{s} = \sqrt{\frac{s_{y,k}}{s_{d,k}}\left( {\left( {1 - \left( \frac{\omega_{d}}{\omega_{n}} \right)^{2}} \right)^{2} + \left( {2\zeta \frac{\omega_{d}}{\omega_{n}}} \right)^{2}} \right)}$

Still referring to FIG. 6, process 600 is shown to include operating aplant using an ESC control technique based on the ESC parameters (step606). Step 606 may include using the ESC parameters to generate a dithersignal d and using the dither signal d to perturb a control input u forthe plant. Step 606 may include monitoring the output signal y from theplant resulting from the perturbed control input u and extracting aperformance gradient p from the output signal y. For example, step 606may include performing a dither-demodulation process to drive theperformance gradient to zero, as described with reference to FIGS. 3A-5.

Process 600 is shown to include determining whether a threshold number Nof cycles have been completed (step 608). If the threshold number N ofcycles (e.g., N≈10) have not been completed (i.e., the result of step608 is “no”), process 600 may return to step 604 and steps 604-608 maybe repeated. However, if the threshold number N of cycles have beencompleted (i.e., the result of step 608 is “yes”), process 600 mayadvance to step 610. In some embodiments, the threshold number N definesa threshold number of time steps k or a threshold amount of timerelative to a predetermined point in time. For example, thepredetermined point in time may be the beginning of process 600 or thetime at which the dither frequency ω_(d) was most recently updated. Inother embodiments, the threshold number N defines a threshold number ofsamples of the dither signal d and/or output variable y collected sincethe predetermined point in time (e.g., N≈1000 at a sampling rate ofapproximately 100 samples per cycle). In some embodiments, the thresholdnumber N defines a threshold number of times steps 604-608 have beenperformed since the dither frequency ω_(d) was most recently updated.

Still referring to FIG. 6, process 600 is shown to include estimatingthe system bandwidth and the system gain (step 610). Step 610 may beperformed by phase delay estimator 520, bandwidth estimator 524, andgain estimator 526, as described with reference to FIG. 5. For example,step 610 may include estimating the phase delay φ between the dithersignal d and the output signal y using the following equation:

$\varphi = {\cos^{- 1}\left( \frac{u_{1} \cdot u_{2}}{{u_{1}}_{2}{u_{2}}_{2}} \right)}$

where u₁ is the dither signal d, u₂ is the output signal y from theplant, u₁·u₂ is the dot product calculated in step 604, and ∥u₁∥₂ and∥u₂∥₂ are the 2-norms calculated in step 604. In other embodiments, step610 may include estimating the phase delay {circumflex over (φ)}_(k) attime k based on the EWMA values s_(d,k), s_(y,k), and s_(x,k) calculatedin step 604. For example, the phase delay {circumflex over (φ)}_(k) attime k may be estimated using the following equation:

${\hat{\varphi}}_{k} = {\cos^{- 1}\left( \frac{s_{x,k}}{\sqrt{s_{d,k}s_{y,k}}} \right)}$

In some embodiments, step 610 includes using the phase delay φ and thedither signal frequency ω_(d) to calculate the system bandwidth ω_(n)using the following equation.

$\omega_{n} = {- {\omega_{d}\left( {{\cot (\varphi)} - \sqrt{\frac{1}{\sin^{2}(\varphi)}}} \right)}}$

In other embodiments, step 610 may include using the phase delay{circumflex over (φ)}_(k) at time k and the dither signal frequencyω_(d) to estimate the system bandwidth {circumflex over (ω)}_(n,k) attime k using the following equation:

${\hat{\omega}}_{n,k} = {- {\omega_{d}\left( {{\cot \left( {\hat{\varphi}}_{k} \right)} - \sqrt{\frac{1}{\sin^{2}\left( {\hat{\varphi}}_{k} \right)}}} \right)}}$

Step 610 may include estimating the system gain K_(s) using thefollowing equation:

$K_{s} = \sqrt{\frac{s_{y,k}}{s_{d,k}}\left( {\left( {1 - \left( \frac{\omega_{d}}{\omega_{n}} \right)^{2}} \right)^{2} + \left( {2\zeta \frac{\omega_{d}}{\omega_{n}}} \right)^{2}} \right)}$

where s_(y,k) and s_(d,k) are the EWMA values calculated in step 604,ω_(d) is the current value of the dither signal frequency, and ω_(n) isthe estimated system bandwidth (e.g., ω_(n) or {circumflex over(ω)}_(n,k)).

Still referring to FIG. 6, process 600 is shown to include updating thedither frequency ω_(d) using the estimated system bandwidth (step 612).Step 612 may include selecting a new dither frequency ω_(d) using thesystem bandwidth ω_(n) or estimated system bandwidth {circumflex over(ω)}_(n,k) calculated in step 610. It may be desirable to select adither signal frequency ω_(d) that is the same or similar to the systembandwidth ω_(n) to enhance the effect of the dither signal d on theperformance variable y (e.g., by increasing the resonance between dithersignal d and plant 503). For example, step 612 may include selecting anew dither frequency ω_(d) that is equal to the system bandwidth ω_(n)or a multiple of the system bandwidth ω_(n).

The updated dither frequency ω_(d) may be stored in memory and used insubsequent iterations of process 600 to generate the dither signal d. Insome embodiments, the updated dither frequency ω_(d) is used tocalculate an updated value for the system gain K_(s) as previouslydescribed. After updating the dither frequency ω_(d) in step 612,process 600 may return to step 604. Steps 604-612 may be repeatediteratively to automatically update the dither frequency ω_(d), systemgain K_(s), and/or other ESC configuration parameters as conditionschange.

Referring now to FIG. 7, a flowchart of another process 700 forautomatically configuring an extremum-seeking control (ESC) system isshown, according to an exemplary embodiment. Process 700 may beperformed by one or more components of self-configuring extremum-seekingcontroller 501 to automatically determine ESC parameters such as thedither signal frequency ω_(d) and the system gain K_(s). Controller 501may use the ESC parameters to operate a plant (e.g., plant 503) using anextremum-seeking control technique.

Process 700 is shown to include receiving an initial value for a ditherfrequency ω_(d) (step 702). The initial value for the dither frequencyω_(d) may be retrieved from memory, specified by a user, received froman external system or device, automatically determined by controller501, or obtained from any other data source. In some embodiments, theinitial value of the dither frequency ω_(d) is based on knowledge of thesystem. For example, the initial value of the dither frequency ω_(d) maybe determined using a system identification procedure. In otherembodiments, the initial value of the dither frequency ω_(d) is astandard value pre-programmed into controller 501. Advantageously, theinitial value of the dither frequency ω_(d) does not need to be selectedbased on knowledge of the system since the dither frequency ω_(d) isautomatically updated in subsequent steps of process 700.

Process 700 is shown to include generating a dither signal d using thedither frequency ω_(d) (step 704) and perturbing a control input u for aplant using the dither signal d (step 706). The dither signal d may be asinusoidal disturbance of a known frequency (i.e., the dither frequencyω_(d)) and may be applied (e.g., added) to the control input u providedto the plant. The dither signal d may be generated according to dithersignal parameters such as a dither amplitude A_(d) and a ditherfrequency ω_(d). In other words, the dither signal d may have a ditheramplitude A_(d) and a dither frequency ω_(d). Step 706 may includeadding the dither signal d to the output v of an integrator (e.g.,integrator 528) to generate the control input u for the plant (e.g.,u=v+d).

In some embodiments, step 704 includes selecting the dither amplitudeA_(d) such that the input perturbation is no larger than 10% of thetotal input range. In some embodiments, step 704 includes selecting thedither amplitude A_(d) such that the corresponding dithered outputamplitude of the performance variable signal y (i.e., the amplitude ofperturbation caused by dither signal d) is at least twice the noiseamplitude. Step 704 may include selecting the dither frequency ω_(d)based on the natural frequency ω_(n) or natural frequency estimate{circumflex over (ω)}_(n,k) received in step 702 or determined in aprevious iteration of process 700. For example, step 704 may includeselecting the dither frequency ω_(d) to be equal to the naturalfrequency ω_(n) or a multiple of the natural frequency ω_(n). In someembodiments, step 704 includes generating multiple dither signals. Thenumber of dither signals generated in step 704 may depend on the numberof control inputs u provided to the plant. In some embodiments, step 704includes generating a dither signal d for each of the control inputs u.

In some embodiments, step 704 includes calculating an EWMA of the dithersignal d. For example, step 704 may include generating a first EWMAs_(d,k) based on a history of values for the dither signal d. Thenotation s_(d,k) indicates that the variable s_(d,k) represents the EWMAof the dither signal d (or filtered dither signal d_(ƒ)) at time k. Thefirst EWMA s_(d,k) may be generated using the following equation:

$s_{d,k} = {s_{d,{k - 1}} + \frac{d_{k}^{2} - s_{d,{k - 1}}}{\min \left( {k,N_{p}} \right)}}$

where s_(d,k) is the EWMA value at time k, s_(d,k-1) is the previousEWMA value (i.e., at time k−1), d_(k) is the value of the dither signald or filtered dither signal d_(ƒ) at time k, and N_(p) is the durationof the forgetting window. In some embodiments, step 704 includesupdating the EWMA value s_(d,k) each time a new sample of the dithersignal d is received (e.g., at the beginning of each time step k). Insome embodiments, step 704 includes generating a value for N_(p) basedon the dither signal frequency ω_(d) as shown in the following equation:

$N_{p} = \frac{10\left( {2\pi} \right)}{\omega_{d}}$

Still referring to FIG. 7, process 700 is shown to include monitoring anoutput signal y from the plant resulting from the perturbed controlinput u (step 708). The plant uses the control input u to affect acontrolled process (e.g., as a control signal for building equipment)and outputs a performance measure in the form of an output signal y. Theoutput signal y may represent a measured or calculated variable that thecontroller seeks to optimize using an extremum-seeking control process.Optimizing output signal y may include minimizing y, maximizing y,controlling y to achieve a setpoint, or otherwise regulating the valueof output signal y by adjusting the value of control input u. Step 708may include observing the value of output signal y over time and storingthe value of output signal y in memory of the controller.

In some embodiments, step 708 includes generating a second EWMA s_(y,k)based on a history of values for the output signal y. The notations_(y,k) indicates that the variable s_(y,k) represents the EWMA of theoutput signal y (or the filtered output signal y_(ƒ)) at time k. Thesecond EWMA s_(y,k) may be generated using the following equation:

$s_{y,k} = {s_{y,{k - 1}} + \frac{y_{k}^{2} - s_{y,{k - 1}}}{\min \left( {k,N_{p}} \right)}}$

where s_(y,k) is the EWMA value at time k, s_(y,k-1) is the previousEWMA value (i.e., at time k−1), y_(k) is the value of the output signaly or the filtered output signal y_(ƒ) at time k, and N_(p) is theduration of the forgetting window.

In some embodiments, step 708 includes generating a third EWMA s_(x,k)based on a product of the output signal y (or the filtered output signaly_(ƒ)) and the dither signal d (or the filtered dither signal d_(ƒ)).The third EWMA s_(x,k) may be generated using the following equation:

$s_{x,k} = {s_{x,{k - 1}} + \frac{{d_{k}y_{k}} - s_{x,{k - 1}}}{\min \left( {k,N_{p}} \right)}}$

where s_(x,k) is the EWMA value at time k, s_(x,k-1) is the previousEWMA value (i.e., at time k−1), d_(k) is the value of the dither signald or filtered dither signal d_(ƒ) at time k, y_(k) is the value of theoutput signal y or the filtered output signal y_(ƒ) at time k, and N_(p)is the duration of the forgetting window. In some embodiments, step 708includes updating the EWMA values s_(y,k) and s_(x,k) each time a newsample of the output signal y and the dither signal d is received (e.g.,at the beginning of each time step k).

Still referring to FIG. 7, process 700 is shown to include calculating aphase delay φ between the output signal y and the dither signal d (step710). Step 710 may be performed by phase delay estimator 520, asdescribed with reference to FIG. 5. In some embodiments, step 710includes estimating the phase delay φ using the original or filteredvalues for the output signal y and the dither signal d. For example,step 710 may include estimating the phase delay φ using the followingproperty of vectors:

$\varphi = {\cos^{- 1}\left( \frac{u_{1} \cdot u_{2}}{{u_{1}}_{2}{u_{2}}_{2}} \right)}$

where u₁ and u₂ are equal length vectors representing the dither signald and the output signal y, respectively, u₁·u₂ is the dot product of thetwo vectors, and the notation ∥ ∥₂ refers to the 2-norm. The dot productu₁·u₂ can be calculated using the following equation:

${u_{1} \cdot u_{2}} = {\sum\limits_{k = 1}^{N}\; {u_{1,k}u_{2,k}}}$

and the 2-norms can be calculated using the following equations:

${u_{1}}_{2} = {{\sqrt{\sum\limits_{k = 1}^{N}\; u_{1,k}^{2}}\mspace{31mu} {u_{2}}_{2}} = \sqrt{\sum\limits_{k = 1}^{N}\; u_{2,k}^{2}}}$

Step 710 may include substituting the dither signal d (or the filtereddither signal d_(ƒ)) for u₁ and the output signal y (or the filteredoutput signal y_(ƒ)) for u₂ to estimate the phase delay between thedither signal d and the output signal y.

In other embodiments, step 710 includes estimating the phase delay φusing the EWMA values s_(d,k), s_(y,k), and s_(x,k). For example, step710 may include estimating the phase delay {circumflex over (φ)}_(k) attime k using the following equation:

${\hat{\varphi}}_{k} = {\cos^{- 1}\left( \frac{s_{x,k}}{\sqrt{s_{d,k}s_{y,k}}} \right)}$

The phase delay φ and/or {circumflex over (φ)}_(k) may be stored inmemory for use in subsequent steps of process 700.

Still referring to FIG. 7, process 700 is shown to include estimatingthe system bandwidth ω_(n) and the system gain K_(s) using the estimatedphase delay φ (step 712). Step 712 may be performed by bandwidthestimator 524 and gain estimator 526, as described with reference toFIG. 5. In some embodiments, step 712 includes using the phase delay φand the dither signal frequency ω_(d) to calculate the system bandwidthω_(n) using the following equation.

$\omega_{n} = {- {\omega_{d}\left( {{\cot (\varphi)} - \sqrt{\frac{1}{\sin^{2}(\varphi)}}} \right)}}$

In other embodiments, step 712 may include using the phase delay{circumflex over (φ)}_(k) at time k and the dither signal frequencyω_(d) to estimate the system bandwidth {circumflex over (ω)}_(n,k) attime k using the following equation:

${\hat{\omega}}_{n,k} = {- {\omega_{d}\left( {{\cot \left( {\hat{\varphi}}_{k} \right)} - \sqrt{\frac{1}{\sin^{2}\left( {\hat{\varphi}}_{k} \right)}}} \right)}}$

Step 712 may include estimating the system gain K_(s) using thefollowing equation:

$K_{s} = \sqrt{\frac{s_{y,k}}{s_{d,k}}\left( {\left( {1 - \left( \frac{\omega_{d}}{\omega_{n}} \right)^{2}} \right)^{2} + \left( {2\zeta \frac{\omega_{d}}{\omega_{n}}} \right)^{2}} \right)}$

where s_(y,k) and s_(d,k) are the EWMA values calculated in steps704-708, ω_(d) is the current value of the dither signal frequency, andω_(n) is the estimated system bandwidth (e.g., ω_(n) or {circumflex over(ω)}_(n,k)).

Still referring to FIG. 7, process 700 is shown to include updating thedither frequency ω_(d) using the estimated system bandwidth (step 714).Step 714 may include selecting a new dither frequency ω_(d) using thesystem bandwidth ω_(n) or estimated system bandwidth {circumflex over(ω)}_(n,k) calculated in step 712. It may be desirable to select adither signal frequency ω_(d) that is the same or similar to the systembandwidth ω_(n) to enhance the effect of the dither signal d on theperformance variable y (e.g., by increasing the resonance between dithersignal d and plant 503). For example, step 714 may include selecting anew dither frequency ω_(d) that is equal to the system bandwidth ω_(n)or a multiple of the system bandwidth ω_(n).

The updated dither frequency ω_(d) may be stored in memory and used insubsequent iterations of process 700 to generate the dither signal d. Insome embodiments, the updated dither frequency ω_(d) is used tocalculate an updated value for the system gain K_(s) as previouslydescribed. After updating the dither frequency ω_(d) in step 714,process 700 may return to step 704. Steps 704-714 may be repeatediteratively to automatically update the dither frequency ω_(d), systemgain K_(s), and/or other ESC configuration parameters as conditionschange.

Example Test Results

Referring now to FIGS. 8-9, several graphs 800, 850, and 900illustrating the self-configuration of extremum-seeking controlparameters provided by the present invention are shown, according to anexemplary embodiment. The systems and methods of the present inventionwere tested using a simulated plant having the following second-ordertransfer function:

${G(s)} = \frac{K_{s}\omega_{n}^{2}}{s^{2} + {2{\zeta\omega}_{n}s} + \omega_{n}^{2}}$

where ζ is the damping coefficient, ω_(n) is the natural frequency ofthe plant, and K_(s) is the system gain. The damping coefficient ζ wasset to a value of ζ=1. The natural frequency ω_(n) was set to a value of

${\omega_{n} = {\frac{2\pi}{200} \approx {0.031\frac{rad}{s}}}},$

shown in FIG. 8 as line 804. The system gain K_(s) was set to a value ofK_(s)=8, shown in FIG. 8 as line 854.

Referring particularly to FIG. 8, the outputs of the automatic parameterestimation process (i.e., natural frequency ω_(n) and system gain K_(s))are shown as a function of the number of samples of the performancevariable y. Graph 800 illustrates the estimated natural frequency ω_(n)(i.e., the output of bandwidth estimator 524) as a function of thenumber of samples. The estimated natural frequency ω_(n) is shown asline 802. Graph 850 illustrates the estimated system gain K_(s) (i.e.,the output of gain estimator 526) as a function of the number ofsamples. The estimated system gain K_(s) is shown as line 852.

As shown in FIG. 8, the estimated parameters ω_(n) and K_(s) convergequickly to their actual values after an initialization period (e.g., athreshold number of cycles) has passed. For example, the estimatednatural frequency ω_(n) converges quickly from its initial estimate of

$\omega_{n} \approx {0.0628\frac{rad}{s}}$

to the actual value of

$\omega_{n} \approx {0.031{\frac{rad}{s}.}}$

Similarly, the estimated system gain K_(s) converges quickly from itsinitial estimate of K_(s)=0 to the actual value of K_(s)=8. Oscillationscan be seen in the estimated parameters, but such oscillations are smallrelative to the estimated values.

Referring now to FIG. 9, a graph 900 illustrating the dither signal 902(i.e., dither signal d) as a function of the number of samples is shown,according to an exemplary embodiment. The dither signal 902 may be asinusoidal signal having a frequency of oscillation known as the ditherfrequency ω_(d), which accounts for the oscillatory characteristics ofdither signal 902. As previously described, the dither frequency ω_(d)may be dynamically updated based on the estimated natural frequencyω_(n). For example, between 0 samples and 1000 samples, the dithersignal 902 is shown having a relatively quick frequency of oscillation,corresponding to the initial natural frequency estimate of

$\omega_{n} \approx {0.0628{\frac{rad}{s}.}}$

After the initialization period has passed (i.e., at approximately 1000samples), the natural frequency estimate is updated to

${\omega_{n} \approx {0.031\frac{rad}{s}}},$

as shown in graph 800. In response to updating the natural frequencyestimate ω_(n), the dither frequency ω_(d) is adjusted based on the newnatural frequency estimate ω_(n). For example, the oscillatory frequencyof the dither signal 902 is shown decreasing to a relatively slowerfrequency of oscillation at approximately 1000 samples to match the newnatural frequency estimate ω_(n). The DC value of the dither signal 902may also be updated using an extremum-seeking control technique in orderto drive the gradient of the performance variable y to zero, aspreviously described. For example, the DC value of the dither signal 902is shown increasing from an initial value of 0 to a new value of 1 atapproximately 1000 samples.

Advantageously, the systems and methods of the present invention may beused to automatically identify the dither frequency ω_(d) and the systemgain K_(s) using the dither signal to perform system identification.This allows the present invention to be integrated within existingextremum-seeking control systems to allow such systems to beautomatically configured without needing to open the loop or employmanual tests. Simulation results have demonstrated the effectiveness ofthe present invention to quickly and accurately identify the systembandwidth ω_(n) and the system gain K_(s), which can be used to adjustthe characteristics of the dither signal d and/or the control input uused to control the plant.

Configuration of Exemplary Embodiments

The construction and arrangement of the systems and methods as shown inthe various exemplary embodiments are illustrative only. Although only afew embodiments have been described in detail in this disclosure, manymodifications are possible (e.g., variations in sizes, dimensions,structures, shapes and proportions of the various elements, values ofparameters, mounting arrangements, use of materials, colors,orientations, etc.). For example, the position of elements may bereversed or otherwise varied and the nature or number of discreteelements or positions may be altered or varied. Accordingly, all suchmodifications are intended to be included within the scope of thepresent disclosure. The order or sequence of any process or method stepsmay be varied or re-sequenced according to alternative embodiments.Other substitutions, modifications, changes, and omissions may be madein the design, operating conditions and arrangement of the exemplaryembodiments without departing from the scope of the present disclosure.

The present disclosure contemplates methods, systems and programproducts on any machine-readable media for accomplishing variousoperations. The embodiments of the present disclosure may be implementedusing existing computer processors, or by a special purpose computerprocessor for an appropriate system, incorporated for this or anotherpurpose, or by a hardwired system. Embodiments within the scope of thepresent disclosure include program products comprising machine-readablemedia for carrying or having machine-executable instructions or datastructures stored thereon. Such machine-readable media can be anyavailable media that can be accessed by a general purpose or specialpurpose computer or other machine with a processor. By way of example,such machine-readable media can comprise RAM, ROM, EPROM, EEPROM, CD-ROMor other optical disk storage, magnetic disk storage or other magneticstorage devices, or any other medium which can be used to carry or storedesired program code in the form of machine-executable instructions ordata structures and which can be accessed by a general purpose orspecial purpose computer or other machine with a processor. Combinationsof the above are also included within the scope of machine-readablemedia. Machine-executable instructions include, for example,instructions and data which cause a general purpose computer, specialpurpose computer, or special purpose processing machines to perform acertain function or group of functions.

Although the figures show a specific order of method steps, the order ofthe steps may differ from what is depicted. Also two or more steps maybe performed concurrently or with partial concurrence. Such variationwill depend on the software and hardware systems chosen and on designerchoice. All such variations are within the scope of the disclosure.Likewise, software implementations could be accomplished with standardprogramming techniques with rule based logic and other logic toaccomplish the various connection steps, processing steps, comparisonsteps and decision steps.

What is claimed is:
 1. A self-configuring extremum-seeking controllercomprising: a dither signal generator that identifies a stored ditherfrequency, generates a dither signal having the stored dither frequency,and uses the dither signal to perturb a control input for a plant; acommunications interface that provides the perturbed control input tothe plant and receives an output signal from the plant resulting fromthe perturbed control input; a phase delay estimator that estimates aphase delay between the output signal and the dither signal; and abandwidth estimator that estimates a bandwidth of the plant based on theestimated phase delay; wherein the dither signal generator updates thestored dither frequency based on the estimated bandwidth.
 2. Thecontroller of claim 1, wherein the dither signal generator generates anew dither signal having the updated dither frequency and uses the newdither signal to perturb the control input for the plant.
 3. Thecontroller of claim 1, wherein the bandwidth estimator: estimates anatural frequency of the plant based on the estimated phase delay; anduses the estimated natural frequency of the plant as the estimatedbandwidth.
 4. The controller of claim 1, wherein the bandwidth estimatorestimates the bandwidth of the plant as a function of the estimatedphase delay and the stored dither frequency.
 5. The controller of claim1, wherein the phase delay estimator: calculates a dot product of thedither signal and the output signal; calculates 2-norms of the dithersignal and the output signal; and estimates the phase delay as afunction of the dot product and the 2-norms.
 6. The controller of claim1, further comprising an exponentially-weighted moving average (EWMA)calculator that calculates a first EWMA of the dither signal and asecond EWMA of the output signal from the plant; wherein the phase delayestimator estimates the phase delay as a function of the first andsecond EWMAs.
 7. The controller of claim 1, further comprising: a gainestimator that estimates a system gain based on the estimated bandwidthand the stored dither frequency; and an integrator that uses theestimated system gain to scale a step size of a gradient descentprocedure performed by the controller.
 8. The controller of claim 7,further comprising an exponentially-weighted moving average (EWMA)calculator that calculates a first EWMA of the dither signal and asecond EWMA of the output signal from the plant; wherein the gainestimator estimates the system gain as a function of the first andsecond EWMAs.
 9. A method for self-configuring one or moreextremum-seeking control parameters used by an extremum-seekingcontroller to modulate a control input for a plant, the methodcomprising: receiving a stored dither frequency at the extremum-seekingcontroller; generating, by the controller, a dither signal having thestored dither frequency and using the dither signal to perturb thecontrol input for the plant; providing the perturbed control input fromthe controller to the plant and receiving, at the controller, an outputsignal from the plant resulting from the perturbed control input;estimating, by the controller, a phase delay between the output signaland the dither signal; estimating, by the controller, a bandwidth of theplant based on the estimated phase delay; and updating, by thecontroller, the stored dither frequency based on the estimatedbandwidth.
 10. The method of claim 9, further comprising: generating anew dither signal having the updated dither frequency; and using the newdither signal to perturb the control input for the plant.
 11. The methodof claim 9, wherein estimating the bandwidth of the plant comprises:estimating a natural frequency of the plant based on the estimated phasedelay; and using the estimated natural frequency of the plant as theestimated bandwidth.
 12. The method of claim 9, wherein the bandwidth ofthe plant is estimated as a function of the estimated phase delay andthe stored dither frequency.
 13. The method of claim 9, whereinestimating the phase delay between the output signal and the dithersignal comprises: calculating a dot product of the dither signal and theoutput signal; calculating 2-norms of the dither signal and the outputsignal; and estimating the phase delay as a function of the dot productand the 2-norms.
 14. The method of claim 9, further comprisingcalculating a first exponentially-weighted moving average (EWMA) of thedither signal and a second EWMA of the output signal from the plant;wherein the phase delay is estimated as a function of the first andsecond EWMAs.
 15. The method of claim 9, further comprising: estimatinga system gain based on the estimated bandwidth and the stored ditherfrequency; and using the estimated system gain to scale a step size of agradient descent procedure performed by the controller.
 16. The methodof claim 9, further comprising calculating a firstexponentially-weighted moving average (EWMA) of the dither signal and asecond EWMA of the output signal from the plant; wherein the system gainis estimated as a function of the first and second EWMAs.
 17. Aself-configuring extremum-seeking controller comprising: a dither signalgenerator that generates a dither signal and uses the dither signal toperturb a control input for a plant; a communications interface thatprovides the perturbed control input to the plant and receives an outputsignal from the plant resulting from the perturbed control input; and aparameter estimator that uses the output signal from the plant inconjunction with the dither signal to generate values for one or moreextremum-seeking control parameters; wherein the dither signal generatoruses the generated values for the extremum-seeking control parameters togenerate the dither signal.
 18. The controller of claim 17, wherein theparameter estimator comprises: a phase delay estimator that estimates aphase delay between the output signal and the dither signal; and abandwidth estimator that estimates a bandwidth of the plant based on theestimated phase delay; wherein the dither signal generator uses theestimated bandwidth to update a stored dither frequency used to generatethe dither signal.
 19. The controller of claim 18, wherein the parameterestimator comprises a gain estimator that estimates a system gain basedon the estimated bandwidth and the stored dither frequency; thecontroller further comprising an integrator that uses the estimatedsystem gain to scale a step size of a gradient descent procedureperformed by the controller.
 20. The controller of claim 19, furthercomprising an exponentially-weighted moving average (EWMA) calculatorthat calculates a first EWMA of the dither signal and a second EWMA ofthe output signal from the plant; wherein at least one of the phasedelay and the system gain is estimated as a function of the first andsecond EWMAs.